arXiv:0810.4875v1 [math.HO] 27 Oct 2008 upre by supported pdhsvso faayi n ierprildffrnileutosi s a If in [1]). equations (see differential the University partial Tokyo and linear at [24] and lectures analysis theory of 1959 of In vision on his Analysis”. papers oped “Algebraic two called published now Sato mathematics an M. of algebraic branch and m new microlocal restrict a to will I related talk, work this his in of But part the some groups. example, quantum describe for in as, bases such crystal discoveries, of important made and butions uiMw,Kyst kmt,Tsi sia ii ao yef Tos myself, Mich`ele Sato, Vergne. and Mikio Tanisaki Oshima, Kaw Toshio Takahiro Okamoto, Kang, Kiyosato Seok-Jin Miwa, Jimbo, suji Michio Brylin th Hotta, Jean-Luc recall Ryoji Barlet, Date, Also Daniel surio whom on. among so collaborators, and many had groups theory representation quantum also systems, but integrable course, ory, of analysis microlocal and h eaainbtenra n ope nlsswsvr strong very was analysis complex a and analysis, real a functional between to theory separation addicted the totally were from people tools time. when that using analysis, at revolutionary constructed was view are of point Sato’s how now realize to X Kashiwara. sacmlxfiainof complexification a is h tr eisln g,i h al ite,we ii aocreate Sato Mikio when sixties, early the in ago, long begins story The c essential given has Masaki approached, he domain the of each In ealta aaiswr oesmn ed fmteais algebr mathematics, of fields many covers work Masaki’s that Recall ti ra oo opeethr oeapcso h oko Ma of work the of aspects some here present to honor great a is It aaiKsiaaadAgbacAnalysis Algebraic and Kashiwara Masaki otiuin ntedmi fmcooa n leri ana algebraic and of overview microlocal brief of a domain is the It in 2007. contributions 27, June Kyoto, birthday, 60th hsppri ae natl ie nhnro aaiKashiwar Masaki of honor in given talk a on based is paper This M ftesheaf the of M O ireSchapira Pierre yefntoson hyperfunctions , X fhlmrhcfntoson functions holomorphic of Abstract 1 M sara nltcmnfl and analytic real a is M r ooooyclasses cohomology are X lysis. ti difficult is It . i main his og the- Hodge , dcomplex nd . tMasaki at existence dwhen nd a’s devel- n sl to yself k,Et- ski, i Tet- ai, alysis. hiyuki Sato’s ontri- eries /60, saki aic d Figure 1: around 1972 and with more recently

Then came Kashiwara’s thesis, dated December 1970 (of course written in Japanese, but translated in English and published by the French Mathe- matical Society [7]) in which he settles the foundations of analytic -module theory and obtains almost all basic results of the theory (compare withD [13]). With -module theory (also constructed independently in the algebraic set- tings byD J. Bernstein [3]), one finally has the tools to treat general systems of linear partial differential equations, as opposed to one equation with one unknown, or to some very particular overdetermined systems. In particular, Kashiwara succeeds in formulating (and solving, but the difficult problem is the functorial formulation) the Cauchy problem for -modules, obtaining D what is now called the Cauchy-Kowalesky-Kashiwara theorem. After the first revolution of hyperfunction theory, Sato made a second one, ten years later, by creating microlocal analysis, a way to analyse objects of a manifold X in the cotangent bundle T ∗X. With Kashiwara and Kawai, they wrote a long paper [25], quoted everywhere as SKK, whose influence has been considerable during the whole seventies among the analysts (and not only the analysts), although very few of them even tried to read the paper. The SKK paper contains Sato’s construction of the sheaf M of mi- C crofunctions, and as a byproduct, the definition of the wave front set. This is essentially what the analysts, led by H¨ormander, remember of this theory (see [6]). But, to my opinion, this is certainly not the only key point of

2 the SKK paper. Another essential fact is that all constructions are made functorially. For example, microfunctions are obtained by first constructing the microlocalization functor µM , and then applying it to the sheaf X of holomorphic functions on a complex manifold X. When you take forOM a real analytic manifold of whom X is a complexification, you get the sheaf M ∗ C (living on TM X, the cornormal bundle to M in X), but if you replace the embedding M ֒ X by the diagonal embedding ∆ ֒ X X, then you get → ∗ → ×∗ the sheaf of microdifferential operators (on T∆(X X) T X) whose theory was developed by Kashiwara and Kawai. With this× approach,≃ you can adapt the six Grothendieck operations to Analysis and obtain a completely new point of view to classical problems (e.g. the Fourier-Sato transformation). Moreover, the SKK paper contains at least two fundamental and ex- tremely deep results, first the involutivity of characteristics, second the struc- ture of systems of microdifferential equations at the generic points of the characteristic variety. More precisely, let X be a complex manifold and let X be the sheaf of rings of microdifferential operators (a kind of localization E of the sheaf X of differential operators). A microdifferential system on D ∗ M an open subset U of T X is a coherent X U -module. Then E | the support char( ) of , also called its characteristic variety, is a • M M closed complex analytic involutive (that is, co-isotropic) subset of U. Of course, the involutivity theorem has a longer history, including the previous work of Guillemin-Quillen-Sternberg [5], and culminating with the purely algebraic proof of Gabber [4]. At generic points of char( ), (after using complex quantized contact • M transformations and infinite order microdifferential operators) is isomorphic to a partial de Rham system: M

∂xi u =0, (i =1,...,p). In the real case, is isomorphic to a mixture of de Rham, Dolbeault M and Hans Lewy systems:

∂x u =0, (i =1,...,p)  i (∂y + √ 1∂y )u =0, (j =1,...,q)  j − j+1 (∂tk + √ 1tk∂tk+1 )u =0, (k =1,...,r).  − From 1970 to 1980, Kashiwara solved almost all fundamental questions of -module theory, proving in particular the rationality of the zeroes of D 3 Figure 2: Masaki Kashiwara, Teresa Monteiro Fernandes and myself around 1975 b-functions [10] and also stating and solving almost all questions related to regular holonomic modules, in particular the Riemann-Hilbert problem. Let us give some details on this part of Kashiwara’s work. In 1975, he proved that the complex F = R om D( , X ) of holomorphic solutions of a holonomic -module has constructibleH M O cohomology and satisfies proper- D M ties which are now translated by saying that F is perverse [9]. Moreover, two years before [8], in 1973, he calculated the local Euler-Poincar´eindex of F using the characteristic cycle associated to and in fact, defining first what M is now called the local Euler obstruction, or equivalently, the intersection of Lagrangian cycles. In 1977 he gave a precise statement of what should be the Riemann-Hilbert correspondence (see [23, p. 287]), the difficulty being to de- fine a suitable class of holonomic -modules, the so-called regular holonomic modules, what he does in the microlocalD setting with Kawai [15] (after re- lated work with Oshima [16]). Then, in 1979, he announces at the 1979/1980 Seminar of Ecole Polytechnique [11] the theorem, giving with some details

4 the main steps of the proof.

(1975) b op b C Dhol( X ) / DC−c( X ) DO kkk ∼ kkk (1977) kkk kkk ? kk (1979−80) b opu D ( X ) holreg D Unfortunately, Masaki did not publish the whole proof before 1984 [12] and some people tried to make his result their own. As everyone knows, if the platonic world of Mathematics is pure and rigorous, these qualities definitely do not apply to the world of mathematicians. Of course, Kashiwara did a lot of other things during this period 1970/80, in particular in the theory of microdifferential equations, but he did not always take the time to publish his results. I remember that I had once in 1978 at Oberwolfach the opportunity to explain to H¨ormander the so-called “watermelon cut theorem” and you can now find it in [6, Th. 9.6.6]. This beautiful theorem asserts in particular that if a hyperfunction u is supported by a half space f 0, then the analytic wave front set of u above the ≥ boundary f = 0 is invariant by the Hamiltonian vector field Hf . After that, essentially from 1980 to 1990, came another period in which I am more involved. Indeed, we developed together the microlocal theory of sheaves (see [18]). To a sheaf F (not necessarily constructible) on a real manifold M, we asso- ciate a closed conic subset SS(F )1 of the cotangent bundle, the microsupport of F , which describes the directions of non propagation of F . The idea of microsupport emerged when, on one side, Masaki noticed that it was possi- ble to recover the characteristic variety of a holonomic -module from the knowledge of the complex of its holomorphic solutionsD by using the van- ishing cycle functor, and when, on my side, I was lead to this notion by remarking that our previous results on propagation for hyperbolic systems was of purely geometrical nature and had almost nothing to do with partial differential equations. One of the main result of the theory asserts that the microsupport of a sheaf F is an involutive subset of the cotangent bundle, but now we are working on real . In case one works on a complex manifold and F is the sheaf of solutions of a coherent -module , the microsupport of F D M 1SS(F ) stands for singular support.

5 Figure 3: Masaki Kashiwara and myself around 1980 coincides with the characteristic variety of . This gives an alternative proof M of the involutivity of characteristics of -modules. Moreover, constructible sheaves on a real manifold are sheaves whoseD microsupport is subanalytic and Lagrangian. This allowed Kashiwara to adapt to the real case the notion of characteristic cycle of a -module and to define the Lagrangian cycle of an R-constructible sheaf. TheD group of Lagrangian cycles is isomorphic to the Grothendieck group of the abelian category of R-constructible sheaves and is also isomorphic to the group of constructible functions. Lagrangian cycles play a basic role in many questions and have been recently extended to higher K-theory by Beilinson [2]. After 90, Kashiwara concentrated mainly on other subjects such as crystal bases, but nevertheless we wrote several papers together. In order to over- come some difficulties related to the microlocalization functor, we were led to generalize the notion of sheaves and to define ind-sheaves [19]. This theory required a lot of technology from category theory, and, as a byproduct, we wrote a whole book on this subject [20]. and Microlocal Analysis are still actively developing in various directions. Let us mention three of them.

6 (i) Recall that Masaki was the first, in 1996, to introduce algebroid stacks in microlocal analysis [14]. Indeed, on a complex contact manifold the sheaf of microdifferential operators does not exist in general and one has to replace sheaves with stacks. Such algebroid stacks are now commonly used on com- plex symplectic manifolds where microdifferential operators are replaced by a variant involving a central parameter ~. Note that Masaki and Raphael Rouquier recently used such rings of operators to make a surprising link with Cherednik algebras [17]. (ii) As a particular case of the theory of ind-sheaves, one gets the theory of usual sheaves on the subanalytic site. Personally, (I am not sure that Masaki shares this point of view) I am convinced that the subanalytic topology is particularly well suited to treat many problems in Analysis and that there are lot of interesting results to be obtained in this direction. (iii) Another very promising direction is the link between the microlocal theory of sheaves and Fukaya’s category. On one side, Nadler and Zaslow [22, 21], adapting the construction of Lagrangian cycles, constructed a category equivalent to Fukaya’s category (on cotangent bundles, not on general real symplectic manifolds). On the other side, Tamarkin [26] also constructed a category which should play this role. Tamarkin’s idea is to add a variable t R whose dual variable τ plays the role of the inverse of ~ and to work ∈ “microlocally” with the category of constructible sheaves on X R in the open set τ > 0 of T ∗(X R). × I hope that this very× sketchy panorama of almost fifty years of Algebraic Analysis (perhaps one should now better call it “Functorial Analysis”) will have convinced you of the importance of the theory and of the fact that Masaki plays the main role in it since the early seventies.

References

[1] E. Andronikof, Interview with Mikio Sato, Notices of the AMS, 54 Vol 2, (2007).

[2] S. Beilinson, Topological ε-factors, math.AG/0610055.

[3] J. Bernstein, Modules over a ring of differential operators. Study of fun- damental solutions of equations with constant coefficients, Funct. Anal- ysis Appl. 5(1971) 89–101.

7 [4] O. Gabber, The integrability of the characteristic variety, Amer. Journ. Math. 103 (1981) 445–468.

[5] V. Guillemin, D. Quillen and S. Sternberg, The integrability of charac- teristics, Comm. Pure and Appl. Math. 23 (1970) 39–77.

[6] L. H¨ormander, The analysis of partial differential operators I, Grundlehren der Mathematischen Wissenschaften 256 Springer-Verlag, (1983).

[7] M. Kashiwara, Algebraic study of systems of partial differential equa- tions, Tokyo 1970 (in Japanese), M´em. Soc. Math. France 63 (1995) (translated by A. D’Agnolo and J-P.Schneiders).

[8] Index theorem for a maximally overdetermined system of linear differential equations, Proc. Japan Acad. 49 (1973) 803–804.

[9] On the maximally overdetermined system of linear differential equations I, Publ. Res. Inst. Math. Sci. 10 (1974/75) 563–579.

[10] B-functions and holonomic systems. Rationality of roots of B- functions, Invent. Math. 38 (1976/77) 33–53.

[11] Faisceaux constructibles et syst`emes holonˆomes d’´equations aux d´eriv´ees partielles lin´eaires `a points singuliers r´eguliers, S´eminaire Goulaouic-Schwartz, 1979–1980 (French), Exp. No. 19 Ecole´ Polytech., Palaiseau, 1980.

[12] The Riemann Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci. 26 (1984) 319–365.

[13] -modules and microlocal calculus, Translations of Mathematical Monographs,D 217 American Math. Soc. (2003).

[14] Quantization of contact manifolds, Publ. RIMS, Kyoto Univ. 32 (1996) 1–5.

[15] M. Kashiwara and T. Kawai, On holonomic systems of microdifferential equations III. Systems with regular singularities, Publ. Res. Inst. Math. Sci. 17 (1981) 813–979.

8 [16] M. Kashiwara and T. Oshima, Systems of differential equations with regular singularities and their boundary value problems, Ann. of Math. 106 (1977) 145–200. [17] M. Kashiwara and R. Rouquier Microlocalization of rational Cherednik algebras, arXiv:math.RT/0705.1245, to appear. [18] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften 292 Springer-Verlag, (1990). [19] Ind-sheaves, Ast´erisque Soc. Math. France 271 (2001). [20] Categories and sheaves, Grundlehren der Mathematischen Wis- senschaften 332 Springer-Verlag, (2006). [21] D. Nadler, Microlocal branes are constructible sheaves, arXiv:math/0612399. [22] D. Nadler and E. Zaslow, Constructible sheaves and the Fukaya Cate- gory, arXiv:math/0604379. To appear in J. Amer. Math. Soc. [23] J-P. Ramis, Additif II `a“variations sur le th`eme GAGA”, Lecture Notes Math. 694 Springer (1978). [24] M. Sato, Theory of hyperfunctions I & II, Journ. Fac. Sci. Univ. Tokyo, 8 (1959–1960)139–193 487–436. [25] M. Sato, T. Kawai, and M. Kashiwara, Hyperfunctions and pseudo- differential equations. In Komatsu (ed.), Hyperfunctions and pseudo- differential equations, Proceedings Katata 1971, Lecture Notes in Math- ematics Springer 287 (1973) 265–529. [26] D. Tamarkin, Microlocal condition for non-displaceablility, arXiv:0809.1584.

Pierre Schapira Institut de Math´ematiques Universit´ePierre et Marie Curie 175, rue du Chevaleret, 75013 Paris, France e-mail: [email protected] http://www.math.jussieu.fr/∼schapira/

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